Question

How many terms must be added in an arithmetic sequence whose first term is 6 and whose common difference is 5 to obtain a sum of 5220?

Answer 1

Hint: According to the given question, we have to find the terms that must be added in an arithmetic sequence whose first term is 6 and whose common difference is 5 to obtain a sum of 5220.
So, first of all we have to use the sum formula of arithmetic sequence that is mentioned below.
$ \Rightarrow {S_n} = \dfrac{n}{2}\left\{ {2a + \left( {n - 1} \right)d} \right\}......................(A)$
Now, we have to put the values of a and d in the formula (A) that is mentioned above and make the quadratic equation in terms of n.
Now, we have to find the roots of the quadratic equation with the help of the formula that is mentioned below,
$ \Rightarrow \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}..................................(B)$

Complete step by step solution:
Step 1: First of all we have to put the value of the first term of an AP which is 6, the common difference of that AP which is 5 and sum of all terms of that AP which is 5220 in formula (A) that is mentioned in the solution hint.
$ \Rightarrow 5220 = \dfrac{n}{2}\left\{ {2 \times 6 + \left( {n - 1} \right) \times 5} \right\}$
Step 2: Now, we have to solve the above expression which is obtain in the solution step 1,
$ \Rightarrow 5220 = \dfrac{n}{2}\left\{ {12 + 5n - 5} \right\}$
Now, we have to multiply by 2 on the both sides, we get,
$ \Rightarrow 5220 \times 2 = 2 \times \dfrac{n}{2}\left\{ {12 + 5n - 5} \right\}$
Now, we have to solve the above expression
$ \Rightarrow 10440 = n\left\{ {7 + 5n} \right\}$
$
   \Rightarrow 10440 = 5{n^2} + 7n \\
   \Rightarrow 5{n^2} + 7n - 10440 = 0 \\
 $
Step 3: Now, we have to find the roots of the quadratic equation which is obtained in the solution step 2 with the help of the formula (B) that is mentioned in the solution hint.
$ \Rightarrow \dfrac{{ - 7 \pm \sqrt {{{\left( 7 \right)}^2} - 4 \times 5 \times \left( { - 10440} \right)} }}{{2 \times 5}}$
Now, we have to solve the above expression
$
   \Rightarrow \dfrac{{ - 7 \pm \sqrt {49 + 208800} }}{{10}} \\
   \Rightarrow \dfrac{{ - 7 \pm \sqrt {208849} }}{{10}} \\
 $
Now, we have to find$\sqrt {208849} $,
$ \Rightarrow \dfrac{{ - 7 \pm 457}}{{10}}$
Step 4: Now, we have to solve the expression obtain in the solution step 3 and find the values of n,
$ \Rightarrow n = \dfrac{{ - 7 + 457}}{{10}}$and $n = \dfrac{{ - 7 - 457}}{{10}}$
$ \Rightarrow n = 45$and$n = - 46.4$
So, n cannot be negative
$ \Rightarrow n = 45$

Final solution: Hence, 45 terms must be added in an arithmetic sequence whose first term is 6 and whose common difference is 5 to obtain a sum of 5220.

Note:
It is necessary to put the values of the first term of an AP, common difference of that AP and sum of all terms of that AP in formula (A) to obtain the quadratic equation in terms of n.
It is necessary to find the both roots of n with the help of the formula (B) that is mentioned in the solution hint.